Z Transforms for the Embedded System Engineer

Inverse Z-Transforms As long as xn is constrained to be causal (xn = 0 for n z-transform is invertible: There is only one xn having a given z-transform X(z). Inversion of the z-transform (getting xn back from X(z)) is accomplished by recognition: What xn would produce that X(z)? Linearity of the z-transform allows. Develop G(z) in a power series, from which the pi can be identiﬁed as the coeﬃcients of the zi. The coeﬃcients can also be calculated by derivation pi = 1 i! DiG(z) dzi z=0 = 1 i! By inspection: decompose G(z) in parts the inverse transforms of which are known; e.g. The partial fractions 3. By a (path) integral on the. Table of z-transfonns (Table 5.1), where z-transform pairs are tabulated for a variety of signals. To find the inverse z-transform of say, z/(z - y), instead of using the complex integration in Eq. (5.2), we consult the table and find the inverse z-transform of z/(z - y) as ynun. In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system. An important example of the unilateral Z-transform is the probability-generating function, where the component is the probability that a discrete random variable takes the value, and the function is usually written as in terms of =.

by Tim Wescott, Wescott Design Services

(note: For a much more in-depth discussion of the z transform, and other practical uses of control theory, see the bookApplied Control Theory for Embedded Systems.)

The z transform is an essential part of a structured control system design. This paper describes the basics of using the z transform to develop control systems, using the sort of math that is familiar to the accomplished embedded system designer.

This paper talks about closed-loop (or feedback) control systems. This is the case when you are implementing a system like the one in Figure 1. The control software combines the feedback with the command to create the drive signal. The drive signal is amplified and applied to the plant, which changes its behavior. The plant behavior is measured and read by the control software as feedback. Since this all goes in one big circle it is called 'closed loop'.

The z transform is useful, but it only works on certain types of problems. If you can describe your plant and your controller using linear difference equations, and if the coefficients of the equations don't change from sample to sample, then your controller and plant are linear and shift-invariant, and you can use the z transform. If the problem at hand won't allow you to model the plant and controller as linear shift invariant systems you won't be able to use the z transform.

While all real-world plants are nonlinear in some sense, and have parameters that vary with time to some degree, the majority of plants can be usefully modeled with linear time invariant differential equations. Once this is done the effects of sampling the input and updating the output at discrete intervals can be taken into account and a difference equation that describes the plant behavior from the software's point of view can be produced.

Let's say that you've reduced the behavior of the plant to a difference equation that looks like

(1)

where x_n is the output of the plant at sample time n, u_n is the command to the DAC at sample time n, and the various a and b are constants set by the design of the plant, driver and ADC circuits.

You can solve (1) for x_n over time if you are given a specific sequence of u_n using techniques very similar to those used to solve differential equations, but the process is just as tedious and obscure as solving differential equations, and just as error prone and frustrating. Furthermore if you want to investigate what happens when you add feedback to the system you would need to solve many equations like (1) over and over again. The z transform allows you to do both of these things.

Just as the Laplace transform is used to solve linear time-invariant differential equations and to deal with many common feedback control problems using continuous-time control, the z transform is used in sampled-time control to deal with linear shift-invariant difference equations.

Officially, the z transform takes a sequence of numbers x_n and transforms it into an expression X(z) that depends on the variable z but not n. That's the transform part: the problem is transformed from one in the sampled time domain (n), and put it into the z domain. The z transform of x is denoted as Z(x) and defined as

.	(2)

For example, if you have a signal x_n = aⁿ you can plug it into (2) and get

.	(3)

There are some complex rules about the values of z for which (2) is valid. While these rules are important in proving the mathematical solidity of the z transform they don't make much difference in it's practical application and can be safely ignored.

The most commonly used and most important property of the z transformation is that the z transform of a signal that is delayed by one sample time is the z transform of the original signal times z^-1

.	(4)

This seems pretty simplistic, but we'll get a lot of mileage out of it. Table 1 lists some other properties of the z transform (see [1]). If you feel inclined, it's good practice to use (2) or (4) to verify each one of these properties.

Property	Comments
k must be a constant.	(5)
(5) and (6) together imply that the z transform is a linear operation.	(6)
This is called a 'first difference' -- itï¿½s the discrete time equivalent to a differential	(7)
A summation is the discrete time equivalent to an integral.	(8)
This is the unit delay -- it's the fundamental operation in discrete-time analysis.	(9)
This is called the 'Final Value Theorem' and is only valid if all the poles of X(z) are stable.	(10)

Table 1: z Transform Properties.

2.1 An Example

Figure 2 shows a motor and geartrain that we might use in a servo system. We're driving it with a voltage, and reading the position of the output gear with a potentiometer. Let's say that we're going to use it as our plant. If we ignore all the nonlinearities in the system, and any difficulties with the motor performance changing over time, and if we set x = v_p and u = v_i, then the difference equation that describes it might look like

.	(11)

You can take the z transform of (11) without knowing what x_n or u_n are by applying (2) as appropriate (remember that x_n transforms to X(z) and u_n transforms to U(z)). If you do this you get

Z Transform Table